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我需要另一組眼睛來告訴我Burnikel和Ziegler的部門的Eiffel實現有什麼問題,特別是「算法2 - 3n/2n」。埃菲爾功能如下所示。類似「Current」的類型是ARRAYED_LIST [NATURAL_8]。換句話說,該實現使用包含8位值的數字(即分支),因此數字以256爲基數。以下是一個失敗呼叫的手動追蹤。 (對不起,參數非常大,但我無法用較短的值重現此錯誤。)在這種情況下,執行步驟在步驟3b之後。Burnikel和Ziegler算法的Eiffel實現中的錯誤2
這是問題所在。該算法對第5步似乎沒有問題,其餘部分「r」以除數後的數字結束。我相信這個錯誤是在步驟3b中,也許是調用了「假設」提供「Beta^n - 1」值的'ones'。 (也許我不明白乙&個Z「測試^ N」符號
這裏是艾菲爾代碼:
three_by_two_divide (a, a3, b: like Current): TUPLE [quot, rem: like Current]
-- Called by `two_by_one_divide'. It has similar structure as
-- `div_three_halves_by_two_halfs', but the arguments to this
-- function have type {JJ_BIG_NATURAL} instead of like `digit'.
-- See Burnikel & Zieler, "Fast Recursive Division", pp 4-8,
-- Algorithm 2.
require
n_not_odd: b.count >= div_limit and b.count \\ 2 = 0
b_has_2n_digits: b.count = a3.count * 2
a_has_2n_digits: a.count = a3.count * 2
local
n: INTEGER
a1, a2: like Current
b1, b2: like Current
tup: TUPLE [quot, rem: like Current]
q, q1, q2, r, r1: like Current
c, d: like Current
do
n := b.count // 2
-- 1) Split `a'
a1 := new_sub_number (n + 1, a.count, a)
a2 := new_sub_number (1, n.max (1), a)
-- 2) Split `b'.
b1 := new_sub_number (n + 1, b.count, b)
b2 := new_sub_number (1, n.max (1), b)
-- 3) Distinguish cases.
if a1 < b1 then
-- 3a) compute Q = floor ([A1,A2]/B1 with remainder.
if b1.count < div_limit then
tup := school_divide (a, b1)
else
tup := two_by_one_divide (a, b1)
end
q := tup.quot
r1 := tup.rem
else
-- 3b) Q = beta^n - 1 and ...
q := ones (n)
-- ... R1 = [A1,A2] - [B1,0] + [0,B1] = [A1,A2] - QB1.
r1 := a + b1
if n > 1 then
b1.shift_left (n)
else
b1.bit_shift_left (zero_digit.bit_count // 2)
end
r1.subtract (b1)
end
-- 4) D = Q * B2
d := q * b2
-- 5) R1 * B^n + A3 - D. (The paper says "a4".)
r1.shift_left (n)
r := r1 + a3 - d
-- 6) As long as R < 0, repeat
from
until not r.is_negative
loop
r := r + b
q.decrement
end
check
remainder_small_enough: r.count <= b.count
-- because remainder must be less than divisor.
end
Result := [q, r]
ensure
-- n_digit_remainder: Result.rem.count = b.count // 2
quotient_has_correct_count: Result.quot.count <= b.count // 2
end
在跟蹤,箭頭指向一條線,我相信是壞的,但我不知道該怎麼辦這裏是跟蹤:。
three_by_two_divide (a = [227,26,41,95,169,93,135,110],
a3 = [92,164,19,39],
b = [161,167,158,41,164,0,0,0])
n := b.count // 2 = 4
-- 1) Split `a'.
a1 := new_sub_number (n + 1, a.count, a) = [227,26,41,95]
a2 := new_sub_number (1, n.max (1), a) = [169,93,135,110]
-- 2) Split `b'.
b1 := new_sub_number (n + 1, b.count, b) = [161,167,158,41]
b2 := new_sub_number (1, n.max (1), b) = [164,0,0,0]
-- 3b) Q = beta^n -1 and ...
--> q := ones (4) = [255,255,255,255] <-- Is this the error?
-- ... R1 = [A1,A2] - [B1,0] + [0,B1].
r1 := a + b1 = [227,26,41,96,75,5,37,151]
b1.shift_left (n) = [161,167,158,41,0,0,0,0]
r1.subtract (b1) = [65,114,139,55,75,5,37,151]
d := q * b2 = [163,255,255,255,92,0,0,0]
r1.shift_left (n) = [227,25,135,184,172,220,37,151,0,0,0,0] -- too big!
r := r1 + a3 - d -= [227,25,135,184,8,220,37,152,0,164,19,39] -- too big!
我知道這是很長,但任何幫助表示讚賞
謝謝亞歷山大,沒有別名,也沒有全球數據。我想我已經通過避免調用'ones'來解決這個問題。 Burnikel&Ziegler認爲A由B劃分,「讓A =Bβ^ n,得到一個new_a:= A-B並記住商以1開始。此時,A小於B,所以按照B&Z的說法進行。我有另一個問題,但我會在一個新問題中發帖。謝謝。 – jjj
我真的不知道如何將'[65,114,139,55,75,5,37,151]'左移4位數'與'r1.shift_left(n)'給出'[227,25,135,184,172,220,37,151,0,0,0,0 ]'。如果你能解釋這樣的行爲,它將有助於理解實現。 –